Question

Estimate the volume of the solid that lies below the surface z = 4x² - 3y and above the rectangle r = [1, 2] x [0, 3]. Use a Riemann sum with m = n = 2 and choose the sample points to be lower left corners. Round your answer to three decimal places.

Answer 1

To estimate the volume of the solid below the given surface and above the given rectangle, we can use a Riemann sum. By dividing the rectangle into four smaller rectangles and choosing the lower left corners as sample points, we can calculate the volume. The estimated volume is approximately 1.750.

Step-by-step explanation:

To estimate the volume of the solid below the surface z = 4x² - 3y and above the rectangle r = [1, 2] x [0, 3], we can use a Riemann sum. With m = n = 2, we divide the rectangle into four smaller rectangles. We choose the lower left corners of each rectangle as the sample points.

1. Divide the x-interval [1, 2] into two equal subintervals: [1, 1.5] and [1.5, 2].
2. Divide the y-interval [0, 3] into two equal subintervals: [0, 1.5] and [1.5, 3].
3. Calculate the height of each rectangle by evaluating the function at the sample point in the lower left corner of each rectangle: z = 4(1)² - 3(0) = 4 and z = 4(1.5)² - 3(1.5) = 3/4.
4. Calculate the area of each rectangle by multiplying the width and height: 1/2 x 1/2 = 1/4 and 1/2 x 3/2 = 3/4.
5. Add up the volumes of the four rectangles: (1/4)(4) + (1/4)(4) + (1/4)(3/4) + (1/4)(3/4) = 7/4.
6. Round the answer to three decimal places: 7/4 ≈ 1.750.